What is the integral from 0 to h of 1/(2sqrt(hx)) ?

The integral from 0 to h of 1/(2sqrt(hx)) is 1

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integral from 0 to h of 1/(2sqrt(hx))

\int_{0}^{h} \frac{1}{2 h x} d x

1

Solution steps

\int_{0}^{h} \frac{1}{2 h x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int_{0}^{h} \frac{1}{h x} d x

\frac{1}{h x} = \frac{1}{h} \frac{1}{x}

= \frac{1}{2} \cdot \int_{0}^{h} \frac{1}{h} \cdot \frac{1}{x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \frac{1}{h} \cdot \int_{0}^{h} \frac{1}{x} d x

Apply the Power Rule

= \frac{1}{2} \cdot \frac{1}{h} [2 x^{\frac{1}{2}}]_{0}^{h}

Simplify

\frac{1}{2} \cdot \frac{1}{h} [2 x^{\frac{1}{2}}]_{0}^{h} : \frac{[ 2 x ] _{0}^{h}}{2 h}

= \frac{[ 2 x ] _{0}^{h}}{2 h}

Compute the boundaries:

2 h

= \frac{2 h}{2 h}

Simplify

= 1

What is the integral from 0 to h of 1/(2sqrt(hx)) ?
The integral from 0 to h of 1/(2sqrt(hx)) is 1